Curvilinear geometries: regular bases

Disks, annuli, pipes, spheres, balls and spherical shells are discretised with bases that are smooth at the axis and at the poles by construction. No condition is imposed at \(r = 0\), and no term is ever regularised as \(1/(r + \varepsilon)\).

Disk, annulus and pipe

In polar coordinates \((\phi, r)\) each azimuthal mode \(e^{im\phi}\) separates. A smooth scalar on the disk of radius \(R\) has the form

\[ f_m(r) = \rho^{|m|}\, w(\rho^2), \qquad \rho = r/R, \]

with \(w\) a polynomial. A vector is written in spin components \(u_\pm = u_r \pm i u_\phi\), which are smooth when \(u_\pm = \rho^{|m\pm1|} \times\) (polynomial in \(\rho^2\)). Conatus expands the polynomial parts in normalised Jacobi polynomials of \(z = 2\rho^2 - 1\) (a Zernike basis), which stays well conditioned at large \(|m|\) where a monomial-weighted Chebyshev basis does not. Gradient, divergence, curl and Laplacian map these spaces into each other with banded matrices. The disk has one boundary, \(r = R\), so a second-order equation takes one condition there.

On the annulus \(r_i \le r \le r_o\) there is no axis: the radial direction is Chebyshev, the operators are written multiplied by powers of \(r\) (polynomial coefficients, hence banded), and a second-order equation takes one condition on each wall. The pipe is the disk times a periodic axial coordinate.

Sphere, ball and shell

On the sphere, vectors and tensors are expanded in spin-weighted spherical harmonics: the tangent components are taken in the spin basis \(e_\pm = (e_\theta \mp i e_\phi)/\sqrt2\), and a component of spin \(s\) is a series of \({}_sY_\ell^m\), regular at both poles. Surface gradient and divergence raise or lower the spin and act diagonally in \(\ell\).

In the ball the radial dependence of each \((\ell, m)\) is again a power of \(r\) times a Jacobi polynomial in \(2r^2 - 1\), the three-dimensional analogue of the disk, so fields are regular at the centre. In the shell \(r_i \le r \le r_o\) the radial direction is Chebyshev; second-order operators are written in first-order form (the Laplacian as the divergence of a gradient plus a lifted tau term), which keeps the \(\ell = 0\) mode well posed.

Conventions

The resolution of a sphere direction is the number of degrees \(L_{\max} + 1\). Coordinates are ordered \((\phi, r)\) on the disk and annulus, \((\phi, r, z)\) in the pipe, \((\phi, \theta)\) on the sphere and \((\phi, \theta, r)\) in the ball and shell; the native layer prints the bases it chose.

References

  • G. M. Vasil, K. J. Burns, D. Lecoanet, S. Olver, B. P. Brown and J. S. Oishi, Tensor calculus in polar coordinates using Jacobi polynomials, J. Comput. Phys. 325 (2016) 53–73.

  • G. M. Vasil, D. Lecoanet, K. J. Burns, J. S. Oishi and B. P. Brown, Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: mathematical analysis and derivations, J. Comput. Phys. X 3 (2019) 100013; D. Lecoanet et al., Part-II: implementation and examples, ibid. 100012.

  • J. P. Boyd and F. Yu, Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk, J. Comput. Phys. 230 (2011).